Joy Christian wrote:Let me note that for the 4-particle GHZS state the condition E(a, b, c, d) = << ABCD >> = +1 or -1 for some specific settings for all runs and thus even for a single run is similar to the familiar condition E(a, b) = << AB >> = +1 or -1 for the 2-particle EPRB state for some specific settings (i.e., for a = b and a = -b, respectively) for all runs and thus even for a single run. In the latter example, it is the condition of perfect correlation (or perfect anti-correlation), which is predicted by quantum mechanics.

For completeness, let me prove this here for the EPRB case, parallelling the proof below which I provided in response to the counterchallenge to me by Tim Maudlin:

For the EPRB case, let us follow the construction of my 3-sphere model presented in this paper: https://arxiv.org/abs/1405.2355. The proof goes through as follows:

What I want to show is AB = -1 for a = +b and AB = +1 for a = -b even for a single run. It would suffice to prove AB = +1 for a = -b since the case AB = -1 for a = +b follows quite similarly. We start with equations (54) and (55) of the above paper, which define the binary valued functions A = +/-1 and B = +/-1, subject to the conservation of the spin-0 defined in equations (65) and (66). The expectation value E(a, b) = < AB > = - a . b is then derived in equations (67) to (75) of the paper using these functions A and B. In fact, the expectation value (75) or (76) follows from the very construction of the functions A and B in the equations (54) and (55), as a geometrical identity within my 3-sphere model. Therefore we can use this geometrical identity to prove that AB = +1 for a = -b. In fact, for the chosen settings this identity reduces simply to E(a, b) = < AB > = +1. But E(a, b) = < AB > = +1 tells us that the average of the number AB is a constant, and it is equal to +1. This is mathematically possible only if AB = +1 for all runs, for a = -b. But if AB = +1 for all runs, then AB = +1 holds also for any given run. Therefore AB = +1 for any single run, for the chosen settings a = -b. QED.

The traditional interpretation of Bell's theorem is based on the claim that in the actual experiments such as the above the Bell-CHSH inequality with the upper bound of 2 is "violated." If so, then the claimant(s) should be able to provide actual experimental data --- event-by-event --- that violates the Bell-CHSH inequality. In other words, they should be able to provide actual experimental data for which the absolute value of the following sum,

E(a, b) + E(a, b' ) + E(a', b) - E(a', b' ) ,

exceeds the bound of 2, where

E(a, b) = << A(a)B(b) >> ,

E(a, b' ) = << A(a)B(b' ) >> ,

E(a', b) = << A(a' )B(b) >> ,

and

E(a', b' ) = << A(a' )B(b' ) >> .

So this is my new (or perhaps not so new) challenge. Please do not try to insult my intelligence by obfuscating the issue with statistics or probabilities. Just face up the challenge!